introduction statistical thermodynamics · 6 outline rewrite history • atoms first!...
TRANSCRIPT
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Introduction Statistical Thermodynamics
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Molecular Simulations◆ Molecular dynamics:
solve equations of motion
◆ Monte Carlo: importance sampling
r1
MD
MC
r2rn
r1r2rn
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Questions
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Questions• How can we prove that this scheme
generates the desired distribution of configurations?
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Questions• How can we prove that this scheme
generates the desired distribution of configurations?
• Why make a random selection of the particle to be displaced?
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5
Questions• How can we prove that this scheme
generates the desired distribution of configurations?
• Why make a random selection of the particle to be displaced?
• Why do we need to take the old configuration again?
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Questions• How can we prove that this scheme
generates the desired distribution of configurations?
• Why make a random selection of the particle to be displaced?
• Why do we need to take the old configuration again?
• How large should we take: delx?
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Questions• How can we prove that this scheme
generates the desired distribution of configurations?
• Why make a random selection of the particle to be displaced?
• Why do we need to take the old configuration again?
• How large should we take: delx?
What is the desired distribution?
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Outline
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OutlineRewrite History
• Atoms first! Thermodynamics last!
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OutlineRewrite History
• Atoms first! Thermodynamics last!
Thermodynamics• First law: conservation of energy
• Second law: in a closed system entropy increase and takes its maximum value at equilibrium
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6
OutlineRewrite History
• Atoms first! Thermodynamics last!
Thermodynamics• First law: conservation of energy
• Second law: in a closed system entropy increase and takes its maximum value at equilibrium
System at constant temperature and volume• Helmholtz free energy decrease and takes its minimum value
at equilibrium
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6
OutlineRewrite History
• Atoms first! Thermodynamics last!
Thermodynamics• First law: conservation of energy
• Second law: in a closed system entropy increase and takes its maximum value at equilibrium
System at constant temperature and volume• Helmholtz free energy decrease and takes its minimum value
at equilibrium
Other ensembles:• Constant pressure
• grand-canonical ensemble
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Atoms first thermodynamics next
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A box of particles
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A box of particles
We have given the particles an intermolecular potential
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A box of particles
We have given the particles an intermolecular potential
Newton: equations of motion
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A box of particles
We have given the particles an intermolecular potential
Newton: equations of motion
F(r) = -ru(r)
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A box of particles
We have given the particles an intermolecular potential
Newton: equations of motion
F(r) = -ru(r)
md2rdt2
= F(r)
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A box of particles
We have given the particles an intermolecular potential
Newton: equations of motion
F(r) = -ru(r)
Conservation of total energy
md2rdt2
= F(r)
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Phase spaceThermodynamics: N,V,E
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }point in phase space
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }point in phase space
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }point in phase space
ΓN 0( )
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }point in phase space
ΓN 0( ) trajectory: classical mechanics
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }point in phase space
ΓN 0( )ΓN t( )
trajectory: classical mechanics
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }
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Phase spaceThermodynamics: N,V,E
Molecular:
ΓN = r1,r2 ,…,rN ,p1,p2 ,…,pN{ }point in phase space
ΓN 0( )ΓN t( )
trajectory: classical mechanics
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }Why this one?
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All trajectories with the same initial total energy should describe the same thermodynamic state
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All trajectories with the same initial total energy should describe the same thermodynamic state
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }
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All trajectories with the same initial total energy should describe the same thermodynamic state
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }
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All trajectories with the same initial total energy should describe the same thermodynamic state
r1,r2 ,…,rN{ }
p1,p2 ,…,pN{ }These trajectories define a probability density in phase space
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Intermezzo 1: phase rule
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Intermezzo 1: phase rule
• Question: explain the phase rule?
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Intermezzo 1: phase rule
• Question: explain the phase rule?• Phase rule: F=2-P+C
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Intermezzo 1: phase rule
• Question: explain the phase rule?• Phase rule: F=2-P+C
• F: degrees of freedom
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Intermezzo 1: phase rule
• Question: explain the phase rule?• Phase rule: F=2-P+C
• F: degrees of freedom
• P: number of phases
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Intermezzo 1: phase rule
• Question: explain the phase rule?• Phase rule: F=2-P+C
• F: degrees of freedom
• P: number of phases
• C: number of components
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Intermezzo 1: phase rule
• Question: explain the phase rule?• Phase rule: F=2-P+C
• F: degrees of freedom
• P: number of phases
• C: number of components
• Why the 2?
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Making a gas
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
2. Specify the number of particles N
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
2. Specify the number of particles N
3. Give the particles: initial positions initial velocities
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
2. Specify the number of particles N
3. Give the particles: initial positions initial velocities
More we cannot do: Newton takes over!
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
2. Specify the number of particles N
3. Give the particles: initial positions initial velocities
More we cannot do: Newton takes over!System will be at constant:
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
2. Specify the number of particles N
3. Give the particles: initial positions initial velocities
More we cannot do: Newton takes over!System will be at constant: N,V,E
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Making a gasWhat do we need to specify to fully define a thermodynamic system?
1. Specify the volume V
2. Specify the number of particles N
3. Give the particles: initial positions initial velocities
More we cannot do: Newton takes over!System will be at constant: N,V,E
(micro-canonical ensemble)
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What is the force I need to apply to prevent the wall from moving?
Pressure
How much work I do?
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Collision with a wall
Elastic collisions:
Does the energy change?
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Collision with a wall
Elastic collisions:
Does the energy change?
What is the force that we need to apply on the wall?
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Pressure
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Pressure• one particle: 2 m
vx
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Pressure• one particle: 2 m
vx
• # particles: ρ A vx
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Pressure• one particle: 2 m
vx
• # particles: ρ A vx
• 50% is the positive directions: 0.5
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Pressure• one particle: 2 m
vx
• # particles: ρ A vx
• 50% is the positive directions: 0.5• P A = F = ρ A m vx
2
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Pressure• one particle: 2 m
vx
• # particles: ρ A vx
• 50% is the positive directions: 0.5• P A = F = ρ A m vx
2
• Kinetic energy: UK = ½ m v2 = ³⁄₂ kB T• (we define temperature)
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Pressure• one particle: 2 m
vx
• # particles: ρ A vx
• 50% is the positive directions: 0.5• P A = F = ρ A m vx
2
• Kinetic energy: UK = ½ m v2 = ³⁄₂ kB T• (we define temperature)
• Pressure: P V = N kB T
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Experiment (1)
NVE1 NVE2
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Experiment (1)
NVE1 NVE2 E1 > E2
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Experiment (1)
NVE1 NVE2 E1 > E2
What will the moveable wall do?
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Experiment (2)
NVE1 NVE2 E1 > E2
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Experiment (2)
NVE1 NVE2 E1 > E2
Now the wall are heavy molecules
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Experiment (2)
NVE1 NVE2 E1 > E2
What will the moveable wall do?Now the wall are heavy molecules
![Page 67: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/67.jpg)
Newton + atoms
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Newton + atoms
• We have a natural formulation of the first law
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Newton + atoms
• We have a natural formulation of the first law
• We have discovered pressure
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Newton + atoms
• We have a natural formulation of the first law
• We have discovered pressure• We have discovered another
equilibrium properties related to the total energy of the system
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Thermodynamics (classical)
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Experiment
NVE1 NVE2 E1 > E2
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Experiment
NVE1 NVE2 E1 > E2
The wall can move and exchange energy: what determines equilibrium ?
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Classical Thermodynamics
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Classical Thermodynamics
• 1st law of Thermodynamics• Energy is conserved
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Classical Thermodynamics
• 1st law of Thermodynamics• Energy is conserved
• 2nd law of Thermodynamics• Heat spontaneously flows from hot
to cold
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Classical Thermodynamics
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Carnot: Entropy difference between two states:
Classical Thermodynamics
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Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Classical Thermodynamics
![Page 80: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/80.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:
Classical Thermodynamics
![Page 81: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/81.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:
Classical Thermodynamics
![Page 82: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/82.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:�U = Q + W
Classical Thermodynamics
![Page 83: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/83.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:
If we carry out a reversible process, we have for any point along the path
�U = Q + W
Classical Thermodynamics
![Page 84: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/84.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:
If we carry out a reversible process, we have for any point along the path
�U = Q + W
dU = TdS + dW
Classical Thermodynamics
![Page 85: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/85.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:
If we carry out a reversible process, we have for any point along the path
�U = Q + W
If we have work by a expansion of a fluid
dU = TdS + dW
Classical Thermodynamics
![Page 86: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/86.jpg)
Carnot: Entropy difference between two states:
�S = SB � SA =
�B
A
dQrev
T
Using the first law we have:
If we carry out a reversible process, we have for any point along the path
�U = Q + W
If we have work by a expansion of a fluid
dU = TdS + dW
dU = TdS − pdV
Classical Thermodynamics
![Page 87: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/87.jpg)
H L
![Page 88: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/88.jpg)
Let us look at the very initial stage
H L
![Page 89: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/89.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
H L
![Page 90: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/90.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
For system HH L
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Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
dSH = �dq
TH
For system HH L
![Page 92: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/92.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
dSH = �dq
TH
For system H
For system L
H L
![Page 93: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/93.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
dSH = �dq
TH
For system H
For system LdSL =
dq
TL
H L
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Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
Hence, for the total system
dSH = �dq
TH
For system H
For system LdSL =
dq
TL
H L
![Page 95: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/95.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
Hence, for the total system
dSH = �dq
TH
For system H
For system LdSL =
dq
TL
dS = dSL + dSH = dq
�1
TL⇥
1
TH
⇥
H L
![Page 96: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/96.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
Hence, for the total system
dSH = �dq
TH
For system H
For system LdSL =
dq
TL
dS = dSL + dSH = dq
�1
TL⇥
1
TH
⇥
Heat goes from warm to cold: or if dq > 0 then TH > TL
H L
![Page 97: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/97.jpg)
Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
Hence, for the total system
dSH = �dq
TH
For system H
For system LdSL =
dq
TL
dS = dSL + dSH = dq
�1
TL⇥
1
TH
⇥
Heat goes from warm to cold: or if dq > 0 then TH > TL
dS > 0This gives for the entropy change:
H L
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Let us look at the very initial stage
dq is so small that the temperatures of the two systems do not change
Hence, for the total system
dSH = �dq
TH
For system H
For system LdSL =
dq
TL
dS = dSL + dSH = dq
�1
TL⇥
1
TH
⇥
Heat goes from warm to cold: or if dq > 0 then TH > TL
Hence, the entropy increases until the two temperatures are equal
dS > 0This gives for the entropy change:
H L
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Question
![Page 100: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/100.jpg)
Question
• Thermodynamics has a sense of time, but not Newton’s dynamics
• Look at a water atoms in reverse
• Look at a movie in reverse
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Question
• Thermodynamics has a sense of time, but not Newton’s dynamics
• Look at a water atoms in reverse
• Look at a movie in reverse
• When do molecules know about the arrow of time?
![Page 102: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/102.jpg)
Thermodynamics (statistical)
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Statistical Thermodynamics
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Statistical Thermodynamics
Basic assumption
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Statistical Thermodynamics
For an isolated system any microscopic configuration is equally likely
Basic assumption
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Statistical Thermodynamics
For an isolated system any microscopic configuration is equally likely
Basic assumption
Consequence
![Page 107: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/107.jpg)
Statistical Thermodynamics
For an isolated system any microscopic configuration is equally likely
Basic assumption
Consequence
All of statistical thermodynamics and equilibrium thermodynamics
![Page 108: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/108.jpg)
Statistical Thermodynamics
For an isolated system any microscopic configuration is equally likely
Basic assumption
Consequence
All of statistical thermodynamics and equilibrium thermodynamics
... b
ut cl
assic
al
ther
modyn
amics
is bas
ed o
n law
s
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Ideal gas
Let us again make an ideal gas
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Ideal gas
Let us again make an ideal gas
We select:
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Ideal gas
Let us again make an ideal gas
We select: (1) N particles,
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Ideal gas
Let us again make an ideal gas
We select: (1) N particles, (2) Volume V,
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Ideal gas
Let us again make an ideal gas
We select: (1) N particles, (2) Volume V, (3) initial velocities
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Ideal gas
Let us again make an ideal gas
We select: (1) N particles, (2) Volume V, (3) initial velocities + positions
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Ideal gas
Let us again make an ideal gas
We select: (1) N particles, (2) Volume V, (3) initial velocities + positions
This fixes; V/n, U/n
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Ideal gas
Basic assumption
Let us again make an ideal gas
We select: (1) N particles, (2) Volume V, (3) initial velocities + positions
This fixes; V/n, U/n
For an isolated system any microscopic configuration is equally likely
![Page 117: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/117.jpg)
What is the probability to find this configuration?
![Page 118: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/118.jpg)
What is the probability to find this configuration?
The system has the same kinetic energy!!
![Page 119: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/119.jpg)
What is the probability to find this configuration?
The system has the same kinetic energy!!
Our basic assumption must be seriously wrong!
![Page 120: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/120.jpg)
What is the probability to find this configuration?
The system has the same kinetic energy!!
Our basic assumption must be seriously wrong!
... but are we doing the statistics correctly?
![Page 121: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/121.jpg)
Question
• Is it safe to be in this room?
![Page 122: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/122.jpg)
... lets look at our statistics correctly
![Page 123: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/123.jpg)
... lets look at our statistics correctly
![Page 124: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/124.jpg)
... lets look at our statistics correctly
![Page 125: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/125.jpg)
... lets look at our statistics correctly
![Page 126: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/126.jpg)
... lets look at our statistics correctly
![Page 127: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/127.jpg)
... lets look at our statistics correctly
![Page 128: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/128.jpg)
... lets look at our statistics correctly
What is the probability to find this configuration?
![Page 129: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/129.jpg)
... lets look at our statistics correctly
Basic assumption:
What is the probability to find this configuration?
![Page 130: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/130.jpg)
... lets look at our statistics correctly
Basic assumption:
What is the probability to find this configuration?
P =1
total # of configurations
![Page 131: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/131.jpg)
... lets look at our statistics correctly
Basic assumption:
number 1 can be put in M positions, number 2 at M positions, etc
What is the probability to find this configuration?
P =1
total # of configurations
![Page 132: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/132.jpg)
... lets look at our statistics correctly
Basic assumption:
number 1 can be put in M positions, number 2 at M positions, etc
What is the probability to find this configuration?
P =1
total # of configurations
Total number of configurations:
![Page 133: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/133.jpg)
... lets look at our statistics correctly
Basic assumption:
number 1 can be put in M positions, number 2 at M positions, etc
What is the probability to find this configuration?
P =1
total # of configurations
Total number of configurations: MN
![Page 134: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/134.jpg)
... lets look at our statistics correctly
Basic assumption:
number 1 can be put in M positions, number 2 at M positions, etc
What is the probability to find this configuration?
P =1
total # of configurations
Total number of configurations: with MN M =V
dr
![Page 135: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/135.jpg)
... lets look at our statistics correctly
Basic assumption:
number 1 can be put in M positions, number 2 at M positions, etc
What is the probability to find this configuration?
P =1
total # of configurations
Total number of configurations: with MN
the larger the volume of the gas the more configurations
M =V
dr
![Page 136: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/136.jpg)
1 2
3
4
5
6
7
8
9
![Page 137: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/137.jpg)
1 2
3
4
5
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
![Page 138: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/138.jpg)
1 2
3
4
5
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
✓�V
V
◆N
![Page 139: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/139.jpg)
1
2
3
45
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
![Page 140: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/140.jpg)
1
2
3
45
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
✓�V
V
◆N
![Page 141: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/141.jpg)
1
2
3
45
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
![Page 142: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/142.jpg)
1
2
3
45
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
✓�V
V
◆N
![Page 143: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/143.jpg)
1
2
3
45
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
![Page 144: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/144.jpg)
1
2
3
45
What is the probability to find the 9 molecules exactly at these 9 positions?
6
7
8
9
✓�V
V
◆N
![Page 145: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/145.jpg)
What is the probability to find this configuration?
![Page 146: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/146.jpg)
What is the probability to find this configuration?
exactly equal as to any other configuration!!!!!!
![Page 147: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/147.jpg)
What is the probability to find this configuration?
exactly equal as to any other configuration!!!!!!
This is reflecting the microscopic reversibility of Newton’s equations of motion. A microscopic system has no “sense” of the direction of time
![Page 148: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/148.jpg)
What is the probability to find this configuration?
Are we asking the right question?
exactly equal as to any other configuration!!!!!!
This is reflecting the microscopic reversibility of Newton’s equations of motion. A microscopic system has no “sense” of the direction of time
![Page 149: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/149.jpg)
Are we asking the right question?
![Page 150: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/150.jpg)
Are we asking the right question?These are microscopic properties; no irreversibility
![Page 151: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/151.jpg)
Are we asking the right question?These are microscopic properties; no irreversibility
Thermodynamic is about macroscopic properties:
![Page 152: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/152.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 153: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/153.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 154: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/154.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
N P(empty)
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 155: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/155.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
N P(empty)
1 0.5
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 156: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/156.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
N P(empty)
1 0.5
2 0.5 x 0.5
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 157: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/157.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
N P(empty)
1 0.5
2 0.5 x 0.5
3 0.5 x 0.5 x 0.5
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 158: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/158.jpg)
Are we asking the right question?
Measure densities: what is the probability that we have all our N gas particle in the upper half?
N P(empty)
1 0.5
2 0.5 x 0.5
3 0.5 x 0.5 x 0.5
1000 10 -301
These are microscopic properties; no irreversibilityThermodynamic is about macroscopic properties:
![Page 159: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/159.jpg)
Summary
![Page 160: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/160.jpg)
Summary
• On a microscopic level all configurations are equally likely
![Page 161: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/161.jpg)
Summary
• On a microscopic level all configurations are equally likely
• On a macroscopic level; as the number of particles is extremely large, the probability that we have a fluctuation from the average value is extremely low
![Page 162: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/162.jpg)
Summary
• On a microscopic level all configurations are equally likely
• On a macroscopic level; as the number of particles is extremely large, the probability that we have a fluctuation from the average value is extremely low
• Let us quantify these statements
![Page 163: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/163.jpg)
Basic assumptionE1 > E2
NVE1 NVE2
![Page 164: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/164.jpg)
Basic assumptionE1 > E2
Let us look at one of our examples; let us assume that the total system is isolate but heat can flow between 1 and 2.NVE1 NVE2
![Page 165: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/165.jpg)
Basic assumption
All micro states will be equally likely!
E1 > E2
Let us look at one of our examples; let us assume that the total system is isolate but heat can flow between 1 and 2.NVE1 NVE2
![Page 166: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/166.jpg)
Basic assumption
All micro states will be equally likely!
... but the number of micro states that give an particular energy distribution (E1,E-E1) not ...
E1 > E2
Let us look at one of our examples; let us assume that the total system is isolate but heat can flow between 1 and 2.NVE1 NVE2
![Page 167: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/167.jpg)
Basic assumption
All micro states will be equally likely!
... but the number of micro states that give an particular energy distribution (E1,E-E1) not ...
E1 > E2
Let us look at one of our examples; let us assume that the total system is isolate but heat can flow between 1 and 2.NVE1 NVE2
... so, we observe the most likely one ...
![Page 168: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/168.jpg)
In a macroscopic system we will observe the most likely one
![Page 169: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/169.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
![Page 170: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/170.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
The summation only depends on the total energy:
![Page 171: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/171.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
The summation only depends on the total energy:P(E1, E2) = C�N1(E1)�N2(E ⇤ E1)
![Page 172: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/172.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
lnP(E1, E2) = lnC + lnN1(E1) + lnN2(E ⌅ E1)
The summation only depends on the total energy:P(E1, E2) = C�N1(E1)�N2(E ⇤ E1)
![Page 173: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/173.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
lnP(E1, E2) = lnC + lnN1(E1) + lnN2(E ⌅ E1)
The summation only depends on the total energy:
We need to find the maximum
P(E1, E2) = C�N1(E1)�N2(E ⇤ E1)
![Page 174: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/174.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
lnP(E1, E2) = lnC + lnN1(E1) + lnN2(E ⌅ E1)
The summation only depends on the total energy:
We need to find the maximum
P(E1, E2) = C�N1(E1)�N2(E ⇤ E1)
d lnP(E1, E2)
dE1=
d lnN (E1, E2)
dE1= 0
![Page 175: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/175.jpg)
In a macroscopic system we will observe the most likely one
P(E1, E2) =N1(E1)�N2(E ⇤ E1)
�E1=EE1=0 N1(E1)�N2(E ⇤ E1)
lnP(E1, E2) = lnC + lnN1(E1) + lnN2(E ⌅ E1)
The summation only depends on the total energy:
We need to find the maximum
P(E1, E2) = C�N1(E1)�N2(E ⇤ E1)
d lnP(E1, E2)
dE1=
d lnN (E1, E2)
dE1= 0
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
![Page 176: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/176.jpg)
We need to find the maximumd [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
![Page 177: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/177.jpg)
We need to find the maximum
d lnN1(E1)
dE1= ⇤
d lnN2(E ⇤ E1)
dE1
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
![Page 178: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/178.jpg)
We need to find the maximum
As the total energy is constant
d lnN1(E1)
dE1= ⇤
d lnN2(E ⇤ E1)
dE1
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
![Page 179: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/179.jpg)
We need to find the maximum
As the total energy is constant
E2 = E � E1
d lnN1(E1)
dE1= ⇤
d lnN2(E ⇤ E1)
dE1
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
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We need to find the maximum
As the total energy is constant
dE1 = ⇤d(E ⇤ E1) = ⇤dE2
E2 = E � E1
d lnN1(E1)
dE1= ⇤
d lnN2(E ⇤ E1)
dE1
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
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We need to find the maximum
As the total energy is constant
dE1 = ⇤d(E ⇤ E1) = ⇤dE2
Which gives as equilibrium condition:
E2 = E � E1
d lnN1(E1)
dE1= ⇤
d lnN2(E ⇤ E1)
dE1
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
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We need to find the maximum
As the total energy is constant
dE1 = ⇤d(E ⇤ E1) = ⇤dE2
Which gives as equilibrium condition:
E2 = E � E1
d lnN1(E1)
dE1=
d lnN2(E2)
dE2
d lnN1(E1)
dE1= ⇤
d lnN2(E ⇤ E1)
dE1
d [lnN1(E1) + lnN2(E ⌅ E1)]
dE1= 0
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Let us define a property (almost S, but not quite) : S
* = lnN E( )
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Let us define a property (almost S, but not quite) :
Equilibrium if:
S* = lnN E( )
d lnN1 E1( )dE1
=d lnN2 E2( )
dE2
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Let us define a property (almost S, but not quite) :
Equilibrium if:
S* = lnN E( )
d lnN1 E1( )dE1
=d lnN2 E2( )
dE2
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
or
![Page 187: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/187.jpg)
Let us define a property (almost S, but not quite) :
Equilibrium if:
And for the total system:
S* = lnN E( )
d lnN1 E1( )dE1
=d lnN2 E2( )
dE2
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
S* = S1
* + S2*
or
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Let us define a property (almost S, but not quite) :
Equilibrium if:
And for the total system:
For a system at constant energy, volume and number of particles the S* increases until it has reached its maximum value at equilibrium
S* = lnN E( )
d lnN1 E1( )dE1
=d lnN2 E2( )
dE2
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
S* = S1
* + S2*
or
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Let us define a property (almost S, but not quite) :
Equilibrium if:
And for the total system:
For a system at constant energy, volume and number of particles the S* increases until it has reached its maximum value at equilibrium
S* = lnN E( )
d lnN1 E1( )dE1
=d lnN2 E2( )
dE2
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
S* = S1
* + S2*
or
What is this magic property S*?
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Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
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Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
Why is maximizing S* the same as maximizing N?
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Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
Why is maximizing S* the same as maximizing N?The logarithm is a monotonically increasing function.
![Page 193: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/193.jpg)
Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
Why is maximizing S* the same as maximizing N?The logarithm is a monotonically increasing function.
Why else is the logarithm a convenient function?
![Page 194: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/194.jpg)
Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
Why is maximizing S* the same as maximizing N?The logarithm is a monotonically increasing function.
Why else is the logarithm a convenient function?Makes S* additive! Leads to extensivity.
![Page 195: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/195.jpg)
Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
Why is maximizing S* the same as maximizing N?The logarithm is a monotonically increasing function.
Why else is the logarithm a convenient function?Makes S* additive! Leads to extensivity.
Why is S* not quite entropy?
![Page 196: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/196.jpg)
Defined a property S* (that is almost S):
S*(E1,E−E1)= lnℵ(E1,E−E1)= lnℵ1(E1)+ lnℵ2 (E−E1)= S1
*(E1)+ S2*(E−E1)
Why is maximizing S* the same as maximizing N?The logarithm is a monotonically increasing function.
Why else is the logarithm a convenient function?Makes S* additive! Leads to extensivity.
Why is S* not quite entropy?
Units! The logarithm is just a unitless quantity.
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Thermal Equilibrium (Review)
E1 > E2
Isolated system that allows heat flow between 1 and 2.NVE1 NVE2
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Thermal Equilibrium (Review)
E1 > E2
Isolated system that allows heat flow between 1 and 2.NVE1 NVE2
ℵ(E1,E−E1)=ℵ1(E1)iℵ2 (E−E1)
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Thermal Equilibrium (Review)
Number of micro states that give an particular energy distribution (E1,E-E1) is maximized with respect to E1.
E1 > E2
Isolated system that allows heat flow between 1 and 2.NVE1 NVE2
ℵ(E1,E−E1)=ℵ1(E1)iℵ2 (E−E1)
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For a partitioning of E between 1 and 2, the number of accessible states is maximized when:
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
![Page 201: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/201.jpg)
For a partitioning of E between 1 and 2, the number of accessible states is maximized when:
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
What do these partial derivatives relate to?
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For a partitioning of E between 1 and 2, the number of accessible states is maximized when:
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
What do these partial derivatives relate to?
Thermal equilibrium --> Temperature!
![Page 203: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/203.jpg)
For a partitioning of E between 1 and 2, the number of accessible states is maximized when:
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
What do these partial derivatives relate to?
Thermal equilibrium --> Temperature!
dE = TdS-pdV + µidNii=1
M
∑
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For a partitioning of E between 1 and 2, the number of accessible states is maximized when:
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
What do these partial derivatives relate to?
Thermal equilibrium --> Temperature!
dE = TdS-pdV + µidNii=1
M
∑
T= ∂E∂S
⎛⎝⎜
⎞⎠⎟V ,Ni
Temperature
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For a partitioning of E between 1 and 2, the number of accessible states is maximized when:
∂S1*
∂E1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N1 ,V1
=∂S2
*
∂E2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟N2 ,V2
What do these partial derivatives relate to?
Thermal equilibrium --> Temperature!
dE = TdS-pdV + µidNii=1
M
∑
T= ∂E∂S
⎛⎝⎜
⎞⎠⎟V ,Ni
Temperature or1T= ∂S
∂E⎛⎝⎜
⎞⎠⎟V ,Ni
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Summary
• Statistical Mechanics:• basic assumption:
• all microstates are equally likely
• Applied to NVE• Definition of Entropy: S = kB ln Ω
• Equilibrium: equal temperatures
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Question
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How large is Ω for a glass of water?
Question
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How large is Ω for a glass of water?• For macroscopic systems, super-astronomically
large.
Question
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How large is Ω for a glass of water?• For macroscopic systems, super-astronomically
large.
• For instance, for a glass of water at room temperature
Question
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How large is Ω for a glass of water?• For macroscopic systems, super-astronomically
large.
• For instance, for a glass of water at room temperature
Question
![Page 212: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/212.jpg)
How large is Ω for a glass of water?• For macroscopic systems, super-astronomically
large.
• For instance, for a glass of water at room temperature
• Macroscopic deviations from the second law of thermodynamics are not forbidden, but they are extremely unlikely.
Question
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Systems at Constant Temperature (different ensembles)
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The 2nd law
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The 2nd law
Entropy of an isolated system can only increase; until equilibrium were it takes its maximum value
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The 2nd law
Entropy of an isolated system can only increase; until equilibrium were it takes its maximum value
Most systems are at constant temperature and volume or pressure?
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The 2nd law
Entropy of an isolated system can only increase; until equilibrium were it takes its maximum value
Most systems are at constant temperature and volume or pressure?
What is the formulation for these systems?
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Constant T and V
1
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Constant T and V
We have our box 1 and a bath1
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Constant T and V
We have our box 1 and a bath
fixed volume but can exchange energy
1
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Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
fixed volume but can exchange energy
1
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Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
fixed volume but can exchange energy
1
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Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
fixed volume but can exchange energy
dU = dq � pdV = 0
1
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Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
fixed volume but can exchange energy
Second lawdU = dq � pdV = 0
1
![Page 225: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/225.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
fixed volume but can exchange energy
Second law dS � 0
dU = dq � pdV = 0
1
![Page 226: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/226.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
dU = dq � pdV = 0
1
![Page 227: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/227.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:
dU = dq � pdV = 0
1
![Page 228: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/228.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:
dU = dq � pdV = 0
dU1 + dUb = 0
1
![Page 229: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/229.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:
dU = dq � pdV = 0
dU1 + dUb = 0 or
1
![Page 230: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/230.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:
dU = dq � pdV = 0
dU1 + dUb = 0 or dU1 = �dUb
1
![Page 231: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/231.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:The bath is so large that the heat flow does not influence the temperature of the bath + the process is reversible
dU = dq � pdV = 0
dU1 + dUb = 0 or dU1 = �dUb
1
![Page 232: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/232.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:The bath is so large that the heat flow does not influence the temperature of the bath + the process is reversible
dU = dq � pdV = 0
dU1 + dUb = 0 or dU1 = �dUb
2nd law:
1
![Page 233: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/233.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:The bath is so large that the heat flow does not influence the temperature of the bath + the process is reversible
dU = dq � pdV = 0
dU1 + dUb = 0 or dU1 = �dUb
2nd law: dS1 + dSb = dS1 +dUb
T� 0
1
![Page 234: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/234.jpg)
Constant T and V
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant volume and temperature
fixed volume but can exchange energy
Second law dS � 0
1st law:The bath is so large that the heat flow does not influence the temperature of the bath + the process is reversible
dU = dq � pdV = 0
dU1 + dUb = 0 or dU1 = �dUb
2nd law: dS1 + dSb = dS1 +dUb
T� 0
TdS1 � dU1 � 0
1
![Page 235: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/235.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
fixed volume but can exchange energy
1
![Page 236: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/236.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
d(U1 ⇤ TS1) � 0
fixed volume but can exchange energy
1
![Page 237: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/237.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
d(U1 ⇤ TS1) � 0
Let us define the Helmholtz free energy: A
fixed volume but can exchange energy
1
![Page 238: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/238.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
d(U1 ⇤ TS1) � 0
Let us define the Helmholtz free energy: AA � U � TS
fixed volume but can exchange energy
1
![Page 239: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/239.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
d(U1 ⇤ TS1) � 0
Let us define the Helmholtz free energy: AA � U � TS
For box 1 we can write
fixed volume but can exchange energy
1
![Page 240: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/240.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
d(U1 ⇤ TS1) � 0
Let us define the Helmholtz free energy: AA � U � TS
For box 1 we can write dA1 � 0
fixed volume but can exchange energy
1
![Page 241: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/241.jpg)
Constant T and V
Total system is isolated and the volume is constantBox 1: constant volume and temperature 2nd law: TdS1 � dU1 � 0
d(U1 ⇤ TS1) � 0
Let us define the Helmholtz free energy: AA � U � TS
For box 1 we can write dA1 � 0
Hence, for a system at constant temperature and volume the Helmholtz free energy decreases and takes its minimum value at equilibrium
fixed volume but can exchange energy
1
![Page 242: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/242.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
![Page 243: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/243.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
![Page 244: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/244.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
1/kBT
![Page 245: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/245.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
![Page 246: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/246.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
![Page 247: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/247.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
![Page 248: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/248.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
![Page 249: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/249.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
![Page 250: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/250.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Boltzmann distribution
![Page 251: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/251.jpg)
51
Canonical ensembleConsider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Boltzmann distribution
![Page 252: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/252.jpg)
52
ThermodynamicsWhat is the average energy of the system?
![Page 253: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/253.jpg)
52
ThermodynamicsWhat is the average energy of the system?
![Page 254: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/254.jpg)
52
ThermodynamicsWhat is the average energy of the system?
![Page 255: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/255.jpg)
52
ThermodynamicsWhat is the average energy of the system?
![Page 256: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/256.jpg)
52
ThermodynamicsWhat is the average energy of the system?
Compare:
![Page 257: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/257.jpg)
53
ThermodynamicsFirst law of thermodynamics
![Page 258: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/258.jpg)
53
ThermodynamicsFirst law of thermodynamics
Helmholtz Free energy:
![Page 259: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/259.jpg)
53
ThermodynamicsFirst law of thermodynamics
Helmholtz Free energy:
![Page 260: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/260.jpg)
53
ThermodynamicsFirst law of thermodynamics
Helmholtz Free energy:
![Page 261: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/261.jpg)
53
ThermodynamicsFirst law of thermodynamics
Helmholtz Free energy:
![Page 262: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/262.jpg)
53
ThermodynamicsFirst law of thermodynamics
Helmholtz Free energy:
![Page 263: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/263.jpg)
54
What is the average energy of the system?
![Page 264: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/264.jpg)
54
What is the average energy of the system?
![Page 265: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/265.jpg)
54
What is the average energy of the system?
![Page 266: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/266.jpg)
54
What is the average energy of the system?
![Page 267: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/267.jpg)
54
What is the average energy of the system?
Compare:
![Page 268: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/268.jpg)
54
What is the average energy of the system?
Compare: Hence:
![Page 269: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/269.jpg)
We have assumed that we can count states
Atoms?
![Page 270: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/270.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
Atoms?
![Page 271: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/271.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
What to do for classical model such as an ideal gas, hard spheres, Lennard-Jones?
Atoms?
![Page 272: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/272.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
What to do for classical model such as an ideal gas, hard spheres, Lennard-Jones?
Energy is continue: • potential energy • kinetic energy
Atoms?
![Page 273: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/273.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
What to do for classical model such as an ideal gas, hard spheres, Lennard-Jones?
Energy is continue: • potential energy • kinetic energy
Particle in a box:
Atoms?
![Page 274: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/274.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
What to do for classical model such as an ideal gas, hard spheres, Lennard-Jones?
Energy is continue: • potential energy • kinetic energy
Particle in a box:εn =
nh( )28mL2
Atoms?
![Page 275: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/275.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
What to do for classical model such as an ideal gas, hard spheres, Lennard-Jones?
Energy is continue: • potential energy • kinetic energy
Particle in a box:εn =
nh( )28mL2
Atoms?
![Page 276: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/276.jpg)
We have assumed that we can count states
Quantum Mechanics: energy discreet
What to do for classical model such as an ideal gas, hard spheres, Lennard-Jones?
Energy is continue: • potential energy • kinetic energy
Particle in a box:εn =
nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Atoms?
![Page 277: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/277.jpg)
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
![Page 278: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/278.jpg)
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
![Page 279: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/279.jpg)
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
![Page 280: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/280.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
![Page 281: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/281.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
Many levels are occupied: only at very low temperatures or very small volumes one can see quantum effect!
![Page 282: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/282.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
Many levels are occupied: only at very low temperatures or very small volumes one can see quantum effect!
qtranslational = e−
nh( )28mL2kBT
n=1
∞
∑
![Page 283: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/283.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
Many levels are occupied: only at very low temperatures or very small volumes one can see quantum effect!
qtranslational = e−
nh( )28mL2kBT
n=1
∞
∑
qtranslational = e−
nh( )28mL2kBT dn
0
∞
∫
![Page 284: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/284.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
Many levels are occupied: only at very low temperatures or very small volumes one can see quantum effect!
qtranslational = e−
nh( )28mL2kBT
n=1
∞
∑
qtranslational = e−
nh( )28mL2kBT dn
0
∞
∫ qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
12L
![Page 285: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/285.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
Many levels are occupied: only at very low temperatures or very small volumes one can see quantum effect!
qtranslational = e−
nh( )28mL2kBT
n=1
∞
∑
qtranslational = e−
nh( )28mL2kBT dn
0
∞
∫ qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
12L
3D: qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
32V = V
Λ3
![Page 286: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/286.jpg)
Kinetic energy of Ar at room temperature ≈4.14 × 10-21 J
εn =nh( )28mL2
What are the energy levels for Argon in a 1-dimensional box of 1 cm?
Argon: m=40 g/mol=6.63×10-26 kg h=6.63×10-34 J s
εn = 5 ×10−39n2 J( )
Many levels are occupied: only at very low temperatures or very small volumes one can see quantum effect!
qtranslational = e−
nh( )28mL2kBT
n=1
∞
∑
qtranslational = e−
nh( )28mL2kBT dn
0
∞
∫ qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
12L
3D: qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
32V = V
Λ3De Broglie wavelength
![Page 287: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/287.jpg)
Partition function; q = e−Enn=1
∞
∑ = e−En dn∫
![Page 288: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/288.jpg)
Partition function; q = e−Enn=1
∞
∑ = e−En dn∫
Hamiltonian: H =Ukin +Upot =pi2
2mi
+Upot rN( )i∑
![Page 289: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/289.jpg)
Partition function; q = e−Enn=1
∞
∑ = e−En dn∫
Hamiltonian: H =Ukin +Upot =pi2
2mi
+Upot rN( )i∑
Z1,V ,T = C e− p2
2m kBT dp3N∫ e−Up r( )kBT dr3N∫
![Page 290: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/290.jpg)
Partition function; q = e−Enn=1
∞
∑ = e−En dn∫
Hamiltonian: H =Ukin +Upot =pi2
2mi
+Upot rN( )i∑
Z1,V ,T = C e− p2
2m kBT dp3N∫ e−Up r( )kBT dr3N∫
One ideal gas molecule: Up(r)=0
![Page 291: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/291.jpg)
Partition function; q = e−Enn=1
∞
∑ = e−En dn∫
Hamiltonian: H =Ukin +Upot =pi2
2mi
+Upot rN( )i∑
Z1,V ,T = C e− p2
2m kBT dp3N∫ e−Up r( )kBT dr3N∫
One ideal gas molecule: Up(r)=0
Z1,V ,TIG = C e
− p2
2m kBT dp∫ dr∫ = CV 2πmkBT( )32
![Page 292: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/292.jpg)
Partition function;
qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
32V = V
Λ3
q = e−Enn=1
∞
∑ = e−En dn∫
Hamiltonian: H =Ukin +Upot =pi2
2mi
+Upot rN( )i∑
Z1,V ,T = C e− p2
2m kBT dp3N∫ e−Up r( )kBT dr3N∫
One ideal gas molecule: Up(r)=0
Z1,V ,TIG = C e
− p2
2m kBT dp∫ dr∫ = CV 2πmkBT( )32
![Page 293: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/293.jpg)
Partition function;
qtranslational =2πmkBT
h2⎛⎝⎜
⎞⎠⎟
32V = V
Λ3
q = e−Enn=1
∞
∑ = e−En dn∫
Hamiltonian: H =Ukin +Upot =pi2
2mi
+Upot rN( )i∑
Z1,V ,T = C e− p2
2m kBT dp3N∫ e−Up r( )kBT dr3N∫
One ideal gas molecule: Up(r)=0
Z1,V ,TIG = C e
− p2
2m kBT dp∫ dr∫ = CV 2πmkBT( )32
Z1,V ,TIG = CV 2πmkBT( )
32 = V
Λ3
![Page 294: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/294.jpg)
N gas molecules:
ZN ,V ,T =1h3N
e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
![Page 295: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/295.jpg)
WRONG!Particles are
indistinguishable
N gas molecules:
ZN ,V ,T =1h3N
e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
![Page 296: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/296.jpg)
WRONG!Particles are
indistinguishable
N gas molecules:
ZN ,V ,T =1
h3NN!e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
ZN ,V ,T =1h3N
e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
![Page 297: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/297.jpg)
WRONG!Particles are
indistinguishable
N gas molecules:
ZN ,V ,T =1
h3NN!e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
ZN ,V ,T =1h3N
e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
Configurational part of the partition function:
![Page 298: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/298.jpg)
WRONG!Particles are
indistinguishable
N gas molecules:
ZN ,V ,T =1
h3NN!e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
QN ,V ,T =1
Λ3NN!e−U r( )kBT drN∫
ZN ,V ,T =1h3N
e− p2
2m kBT dpN∫ e−U r( )kBT drN∫
Configurational part of the partition function:
![Page 299: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/299.jpg)
Question
• For an ideal gas, calculate:• the partition function
• the pressure
• the energy
• the chemical potential
![Page 300: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/300.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
All thermodynamics follows from the partition function!
![Page 301: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/301.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy:
All thermodynamics follows from the partition function!
![Page 302: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/302.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
All thermodynamics follows from the partition function!
![Page 303: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/303.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
![Page 304: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/304.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
Pressure:
![Page 305: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/305.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
Pressure: p = − ∂F∂V
⎛⎝⎜
⎞⎠⎟ T ,N
= kBTN1V
![Page 306: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/306.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
Pressure: p = − ∂F∂V
⎛⎝⎜
⎞⎠⎟ T ,N
= kBTN1V
Energy:
![Page 307: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/307.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
Pressure: p = − ∂F∂V
⎛⎝⎜
⎞⎠⎟ T ,N
= kBTN1V
Energy: E = ∂F T∂1 T
⎛⎝⎜
⎞⎠⎟V ,N
= 3kBN∂lnΛ∂1 T
⎛⎝⎜
⎞⎠⎟V ,N
![Page 308: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/308.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
Pressure: p = − ∂F∂V
⎛⎝⎜
⎞⎠⎟ T ,N
= kBTN1V
Energy: E = ∂F T∂1 T
⎛⎝⎜
⎞⎠⎟V ,N
= 3kBN∂lnΛ∂1 T
⎛⎝⎜
⎞⎠⎟V ,N
Λ = h2
2πmkBT⎛⎝⎜
⎞⎠⎟
12
![Page 309: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/309.jpg)
ideal gas molecules: QN ,V ,TIG = V N
Λ3NN!
Free energy: FIG = −kBT lnQN ,V ,TIG = kBTN lnΛ3 − ln V N( )⎡⎣ ⎤⎦
FIG = F0 + kBTN lnρ
All thermodynamics follows from the partition function!
Pressure: p = − ∂F∂V
⎛⎝⎜
⎞⎠⎟ T ,N
= kBTN1V
Energy: E = ∂F T∂1 T
⎛⎝⎜
⎞⎠⎟V ,N
= 3kBN∂lnΛ∂1 T
⎛⎝⎜
⎞⎠⎟V ,N
Λ = h2
2πmkBT⎛⎝⎜
⎞⎠⎟
12
E = 32NkBT
![Page 310: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/310.jpg)
61
Chemical potential: µi =∂F∂Ni
⎛⎝⎜
⎞⎠⎟ T ,V ,N j
![Page 311: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/311.jpg)
61
Chemical potential: µi =∂F∂Ni
⎛⎝⎜
⎞⎠⎟ T ,V ,N j
βF = N lnΛ3 + N ln N
V⎛⎝⎜
⎞⎠⎟
![Page 312: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/312.jpg)
61
Chemical potential: µi =∂F∂Ni
⎛⎝⎜
⎞⎠⎟ T ,V ,N j
βF = N lnΛ3 + N ln N
V⎛⎝⎜
⎞⎠⎟
βµ = lnΛ3 + lnρ +1
![Page 313: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/313.jpg)
61
Chemical potential: µi =∂F∂Ni
⎛⎝⎜
⎞⎠⎟ T ,V ,N j
βF = N lnΛ3 + N ln N
V⎛⎝⎜
⎞⎠⎟
βµ = lnΛ3 + lnρ +1
βµIG = βµ0 + lnρ
![Page 314: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/314.jpg)
62
Summary:Canonical ensemble (N,V,T)
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62
Summary:Canonical ensemble (N,V,T)
Partition function:
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62
Summary:Canonical ensemble (N,V,T)
Partition function:
Probability to find a particular configuration
![Page 317: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/317.jpg)
62
Summary:Canonical ensemble (N,V,T)
Partition function:
Probability to find a particular configuration
Free energy
![Page 318: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/318.jpg)
63
Summary:micro-canonical ensemble (N,V,E)
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63
Summary:micro-canonical ensemble (N,V,E)
Partition function:
![Page 320: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/320.jpg)
63
Summary:micro-canonical ensemble (N,V,E)
Partition function:
Probability to find a particular configuration
![Page 321: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/321.jpg)
63
Summary:micro-canonical ensemble (N,V,E)
Partition function:
Probability to find a particular configuration
Free energy
![Page 322: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/322.jpg)
Other Ensemble
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65
Other ensembles?
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65
Other ensembles?In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer …
![Page 325: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/325.jpg)
65
Other ensembles?In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully select an appropriate ensemble.
![Page 326: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/326.jpg)
65
Other ensembles?In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
![Page 327: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/327.jpg)
65
Other ensembles?In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
But for doing this wee need to know the Statistical Thermodynamics of the various ensembles.
![Page 328: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/328.jpg)
65
Other ensembles?In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
But for doing this wee need to know the Statistical Thermodynamics of the various ensembles.
COURSE: MD and MC different
ensembles
![Page 329: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/329.jpg)
65
Other ensembles?In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
But for doing this wee need to know the Statistical Thermodynamics of the various ensembles.
![Page 330: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/330.jpg)
66
Example (1): vapour-liquid equilibrium mixture
composition
T
L
VL+V
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66
Example (1): vapour-liquid equilibrium mixture
Measure the composition of the coexisting vapour and liquid phases if we start with a homogeneous liquid of two different compositions:
composition
T
L
VL+V
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66
Example (1): vapour-liquid equilibrium mixture
Measure the composition of the coexisting vapour and liquid phases if we start with a homogeneous liquid of two different compositions:
• How to mimic this with the N,V,T ensemble?
composition
T
L
VL+V
![Page 333: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/333.jpg)
66
Example (1): vapour-liquid equilibrium mixture
Measure the composition of the coexisting vapour and liquid phases if we start with a homogeneous liquid of two different compositions:
• How to mimic this with the N,V,T ensemble?
• What is a better ensemble?composition
T
L
VL+V
![Page 334: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/334.jpg)
67
Example (2): swelling of clays
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67
Example (2): swelling of clays
Deep in the earth clay layers can swell upon adsorption of water:
• How to mimic this in the N,V,T ensemble?
• What is a better ensemble to use?
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68
Ensembles
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68
Ensembles
• Micro-canonical ensemble: E,V,N
• Canonical ensemble: T,V,N
• Constant pressure ensemble: T,P,N
• Grand-canonical ensemble: T,V,µ
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Constant pressure
![Page 339: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/339.jpg)
fixed N but can exchange energy + volume
1
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We have our box 1 and a bath
fixed N but can exchange energy + volume
1
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We have our box 1 and a bathTotal system is isolated and the volume is constant
fixed N but can exchange energy + volume
1
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We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
fixed N but can exchange energy + volume
1
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We have our box 1 and a bathTotal system is isolated and the volume is constant
First law dU = dq � pdV = 0
fixed N but can exchange energy + volume
1
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We have our box 1 and a bathTotal system is isolated and the volume is constant
First lawSecond law
dU = dq � pdV = 0
fixed N but can exchange energy + volume
1
![Page 345: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/345.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First lawSecond law dS � 0
dU = dq � pdV = 0
fixed N but can exchange energy + volume
1
![Page 346: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/346.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
dU = dq � pdV = 0
fixed N but can exchange energy + volume
1
![Page 347: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/347.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
fixed N but can exchange energy + volume
1
![Page 348: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/348.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
dU1 + dUb = 0
fixed N but can exchange energy + volume
1
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We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
or dU1 + dUb = 0
fixed N but can exchange energy + volume
1
![Page 350: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/350.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
or dU1 = �dUbdU1 + dUb = 0
fixed N but can exchange energy + volume
1
![Page 351: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/351.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
or dU1 = �dUbdU1 + dUb = 0dV1 + dVb = 0
fixed N but can exchange energy + volume
1
![Page 352: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/352.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
or dU1 = �dUbdU1 + dUb = 0dV1 + dVb = 0 or
fixed N but can exchange energy + volume
1
![Page 353: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/353.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
dU = dq � pdV = 0
or dU1 = �dUbdU1 + dUb = 0dV1 + dVb = 0 or dV1 = �dVb
fixed N but can exchange energy + volume
1
![Page 354: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/354.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
The bath is very large and the small changes do not change P or T; in addition the process is reversible
dU = dq � pdV = 0
or dU1 = �dUbdU1 + dUb = 0dV1 + dVb = 0 or dV1 = �dVb
fixed N but can exchange energy + volume
1
![Page 355: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/355.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
The bath is very large and the small changes do not change P or T; in addition the process is reversible
dU = dq � pdV = 0
or dU1 = �dUb
2nd law:
dU1 + dUb = 0dV1 + dVb = 0 or dV1 = �dVb
fixed N but can exchange energy + volume
1
![Page 356: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/356.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
The bath is very large and the small changes do not change P or T; in addition the process is reversible
dU = dq � pdV = 0
or dU1 = �dUb
2nd law:
dU1 + dUb = 0dV1 + dVb = 0 or dV1 = �dVb
dS1 + dSb = dS1 +dUb
T+
p
TdVb � 0
fixed N but can exchange energy + volume
1
![Page 357: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/357.jpg)
We have our box 1 and a bathTotal system is isolated and the volume is constant
First law
Box 1: constant pressure and temperature Second law dS � 0
1st law:
The bath is very large and the small changes do not change P or T; in addition the process is reversible
dU = dq � pdV = 0
or dU1 = �dUb
2nd law:
dU1 + dUb = 0dV1 + dVb = 0 or dV1 = �dVb
TdS1 � dU1 � pdV1 � 0
dS1 + dSb = dS1 +dUb
T+
p
TdVb � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:
d(U1 ⌅ TS1 + pV1) � 0
TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:
Let us define the Gibbs free energy: G
d(U1 ⌅ TS1 + pV1) � 0
TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:
Let us define the Gibbs free energy: G
d(U1 ⌅ TS1 + pV1) � 0
G � U ⇥ TS + pV
TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:
Let us define the Gibbs free energy: GFor box 1 we can write
d(U1 ⌅ TS1 + pV1) � 0
G � U ⇥ TS + pV
TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:
Let us define the Gibbs free energy: GFor box 1 we can write
d(U1 ⌅ TS1 + pV1) � 0
G � U ⇥ TS + pV
dG1 � 0
TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
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Total system is isolated and the volume is constantBox 1: constant pressure and temperature 2nd law:
Let us define the Gibbs free energy: GFor box 1 we can write
Hence, for a system at constant temperature and pressure the Gibbs free energy decreases and takes its minimum value at equilibrium
d(U1 ⌅ TS1 + pV1) � 0
G � U ⇥ TS + pV
dG1 � 0
TdS1 � dU1 � pdV1 � 0
fixed N but can exchange energy + volume
1
![Page 365: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/365.jpg)
N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
![Page 366: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/366.jpg)
N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
![Page 367: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/367.jpg)
N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
The terms in the expansion follow from the connection with Thermodynamics:
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N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
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N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
We have:
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N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µi
TdNi∑We have:
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N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µi
TdNi∑We have:
∂S∂U
⎛⎝⎜
⎞⎠⎟V ,Ni
= 1T
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N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µi
TdNi∑We have:
∂S∂U
⎛⎝⎜
⎞⎠⎟V ,Ni
= 1T
and
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N,P,T ensemble
Consider a small system that can exchange volume and energy with a big reservoir
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µi
TdNi∑We have:
∂S∂U
⎛⎝⎜
⎞⎠⎟V ,Ni
= 1T
∂S∂V
⎛⎝⎜
⎞⎠⎟ E ,Ni
= pT
and
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lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂V
⎛⎝⎜
⎞⎠⎟ E ,N
Vi +!
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lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− Ei
kBT− pkBT
Vi
lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂V
⎛⎝⎜
⎞⎠⎟ E ,N
Vi +!
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lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− Ei
kBT− pkBT
Vi
lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂V
⎛⎝⎜
⎞⎠⎟ E ,N
Vi +!
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Hence, the probability to find Ei,Vi:
lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− Ei
kBT− pkBT
Vi
lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂V
⎛⎝⎜
⎞⎠⎟ E ,N
Vi +!
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Hence, the probability to find Ei,Vi:
lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− Ei
kBT− pkBT
Vi
lnΩ V −Vi ,E − Ei( ) = lnΩ V ,E( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂V
⎛⎝⎜
⎞⎠⎟ E ,N
Vi +!
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Partition function:
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Ensemble average:
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
V =Vj expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Δ N , p,T( ) = −kBT∂ lnΔ∂p
⎛⎝⎜
⎞⎠⎟ T ,N
Ensemble average:
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
V =Vj expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Δ N , p,T( ) = −kBT∂ lnΔ∂p
⎛⎝⎜
⎞⎠⎟ T ,N
Ensemble average:
Thermodynamics
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
V =Vj expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Δ N , p,T( ) = −kBT∂ lnΔ∂p
⎛⎝⎜
⎞⎠⎟ T ,N
Ensemble average:
dG = −SdT +Vdp + µidNi∑Thermodynamics
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
V =Vj expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Δ N , p,T( ) = −kBT∂ lnΔ∂p
⎛⎝⎜
⎞⎠⎟ T ,N
Ensemble average:
dG = −SdT +Vdp + µidNi∑Thermodynamics
V = ∂G∂p
⎛⎝⎜
⎞⎠⎟ T ,N
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
V =Vj expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Δ N , p,T( ) = −kBT∂ lnΔ∂p
⎛⎝⎜
⎞⎠⎟ T ,N
Ensemble average:
dG = −SdT +Vdp + µidNi∑Thermodynamics
V = ∂G∂p
⎛⎝⎜
⎞⎠⎟ T ,NHence:
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Partition function: Δ N ,P,T( ) = expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
V =Vj expi, j∑ − Ei
kBT−pVj
kBT⎡
⎣⎢
⎤
⎦⎥
Δ N , p,T( ) = −kBT∂ lnΔ∂p
⎛⎝⎜
⎞⎠⎟ T ,N
Ensemble average:
dG = −SdT +Vdp + µidNi∑Thermodynamics
V = ∂G∂p
⎛⎝⎜
⎞⎠⎟ T ,NHence:
GkBT
= − lnΔ N , p,T( )
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75
SummaryIn the classical limit, the partition function becomes
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75
SummaryIn the classical limit, the partition function becomes
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75
SummaryIn the classical limit, the partition function becomes
The probability to find a particular configuration:
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grand-canonical ensemble
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Grand-canonical ensemble
Classical• A small system that can exchange heat and
particles with a large bath
Statistical• Taylor expansion of a small reservoir
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Constant T and μ
1
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Constant T and μ
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constant1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst lawdU = TdS − pdV + µdN = 0
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Second lawdU = TdS − pdV + µdN = 0
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Second law dS � 0
dU = TdS − pdV + µdN = 0
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
dU = TdS − pdV + µdN = 0
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
1st law:
dU = TdS − pdV + µdN = 0
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
1st law: or dU1 = �dUbdU1 + dUb = 0
dU = TdS − pdV + µdN = 0
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
1st law: or dU1 = �dUbdU1 + dUb = 0or
dU = TdS − pdV + µdN = 0
dN1 + dNb = 0 dNb = −dN1
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
1st law:
The bath is very large and the small changes do not change μ or T; in addition the process is reversible
or dU1 = �dUbdU1 + dUb = 0or
dU = TdS − pdV + µdN = 0
dN1 + dNb = 0 dNb = −dN1
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
1st law:
The bath is very large and the small changes do not change μ or T; in addition the process is reversible
or dU1 = �dUb
2nd law:
dU1 + dUb = 0or
dU = TdS − pdV + µdN = 0
dN1 + dNb = 0 dNb = −dN1
1
exchange energy and particles
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Constant T and μ
Total system is isolated and the volume is constantFirst law
Box 1: constant chemical potential and temperature
Second law dS � 0
1st law:
The bath is very large and the small changes do not change μ or T; in addition the process is reversible
or dU1 = �dUb
2nd law:
dU1 + dUb = 0or
dU = TdS − pdV + µdN = 0
dN1 + dNb = 0 dNb = −dN1
dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
1
exchange energy and particles
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
For the Gibbs free energy we can write:
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
G ≡U −TS + pVG = µN
For the Gibbs free energy we can write:
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
G ≡U −TS + pVG = µN
For the Gibbs free energy we can write:
− pV =U −TS − µNor
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
G ≡U −TS + pVG = µN
For the Gibbs free energy we can write:
− pV =U −TS − µNor
Giving:
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
G ≡U −TS + pVG = µN
For the Gibbs free energy we can write:
− pV =U −TS − µN
d − pV( ) ≤ 0
or
Giving:
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
G ≡U −TS + pVG = µN
For the Gibbs free energy we can write:
− pV =U −TS − µN
d − pV( ) ≤ 0
or
or d pV( ) ≥ 0Giving:
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dS1 + dSb = dS1 +1TbdUb −
µb
TbdNb ≥ 0
We can express the changes of the bath in terms of properties of the system
dS1 −1TdU1 +
µTdN1 ≥ 0 d TS1 −U1 + µN1( ) ≥ 0d U −TS − µN( ) ≤ 0
G ≡U −TS + pVG = µN
For the Gibbs free energy we can write:
− pV =U −TS − µN
d − pV( ) ≤ 0
or
or
Hence, for a system at constant temperature and chemical potential pV increases and takes its maximum value at equilibrium
d pV( ) ≥ 0Giving:
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80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
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80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
![Page 422: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/422.jpg)
80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
The terms in the expansion follow from the connection with Thermodynamics:
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80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
![Page 424: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/424.jpg)
80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µ
TdN
![Page 425: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/425.jpg)
80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µ
TdN
∂S∂U
⎛⎝⎜
⎞⎠⎟V ,N
= 1T
![Page 426: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/426.jpg)
80
µ,V,T ensemble
Consider a small system that can exchange particles and energy with a big reservoir
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
S = kB lnΩThe terms in the expansion follow from the connection with Thermodynamics:
dS = 1TdU + p
TdV − µ
TdN
∂S∂U
⎛⎝⎜
⎞⎠⎟V ,N
= 1T
∂S∂N
⎛⎝⎜
⎞⎠⎟ E ,V
= − µT
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lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
![Page 428: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/428.jpg)
lnΩ E − Ei ,N − N j( ) = lnΩ E,N( )− Ei
kBT+ µkBT
N j
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
![Page 429: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/429.jpg)
lnΩ E − Ei ,N − N j( ) = lnΩ E,N( )− Ei
kBT+ µkBT
N j
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
lnΩ E − Ei ,N − N j( )
Ω E,N( ) = − Ei
kBT+ µkBT
N j
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Hence, the probability to find Ei,Nj:
lnΩ E − Ei ,N − N j( ) = lnΩ E,N( )− Ei
kBT+ µkBT
N j
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
lnΩ E − Ei ,N − N j( )
Ω E,N( ) = − Ei
kBT+ µkBT
N j
![Page 431: Introduction Statistical Thermodynamics · 6 Outline Rewrite History • Atoms first! Thermodynamics last! Thermodynamics • First law: conservation of energy • Second law: in](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a83cf2891582547592fe4/html5/thumbnails/431.jpg)
Hence, the probability to find Ei,Nj:
lnΩ E − Ei ,N − N j( ) = lnΩ E,N( )− Ei
kBT+ µkBT
N j
lnΩ E − Ei ,N − N j ,( ) = lnΩ E,N( )− ∂ lnΩ
∂E⎛⎝⎜
⎞⎠⎟V ,N
Ei −∂ lnΩ∂N
⎛⎝⎜
⎞⎠⎟ E ,V
N j +!
lnΩ E − Ei ,N − N j( )
Ω E,N( ) = − Ei
kBT+ µkBT
N j
P Ei ,N j( ) = Ω E − Ei ,N − N j( )Ω E − Ek ,N − Nl( )k ,l∑ ∝ exp − Ei
kBT+ µNi
kBT⎡
⎣⎢
⎤
⎦⎥
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82
μ,V,T ensemble (2)In the classical limit, the partition function becomes
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82
μ,V,T ensemble (2)In the classical limit, the partition function becomes
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82
μ,V,T ensemble (2)In the classical limit, the partition function becomes
The probability to find a particular configuration: